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Split Normal Distribution
In probability theory and statistics, the split normal distribution also known as the two-piece normal distribution results from joining at the mode the corresponding halves of two normal distributions with the same mode but different variances. It is claimed by Johnson et al. that this distribution was introduced by Gibbons and Mylroie and by John. But these are two of several independent rediscoveries of the Zweiseitige Gauss'sche Gesetz introduced in the posthumously published ''Kollektivmasslehre'' (1897) of Gustav Theodor Fechner (1801-1887), see Wallis (2014). Another rediscovery has appeared more recently in a finance journal.de Roon, F. and Karehnke, P. (2016). A simple skewed distribution with asset pricing applications. ''Review of Finance'', 2016, 1-29. Definition The split normal distribution arises from merging two opposite halves of two probability density functions (PDFs) of normal distributions in their common mode. The PDF of the split normal distribution is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflation Targeting
In macroeconomics, inflation targeting is a monetary policy where a central bank follows an explicit target for the inflation rate for the medium-term and announces this inflation target to the public. The assumption is that the best that monetary policy can do to support long-term growth of the economy is to maintain price stability, and price stability is achieved by controlling inflation. The central bank uses interest rates as its main short-term monetary instrument. An inflation-targeting central bank will raise or lower interest rates based on above-target or below-target inflation, respectively. The conventional wisdom is that raising interest rates usually cools the economy to rein in inflation; lowering interest rates usually accelerates the economy, thereby boosting inflation. The first three countries to implement fully-fledged inflation targeting were New Zealand, Canada and the United Kingdom in the early 1990s, although Germany had adopted many elements of inflat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflation
In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduction in the purchasing power of money. The opposite of inflation is deflation, a sustained decrease in the general price level of goods and services. The common measure of inflation is the inflation rate, the annualized percentage change in a general price index. As prices do not all increase at the same rate, the consumer price index (CPI) is often used for this purpose. The employment cost index is also used for wages in the United States. Most economists agree that high levels of inflation as well as hyperinflation—which have severely disruptive effects on the real economy—are caused by persistent excessive growth in the money supply. Views on low to moderate rates of inflation are more varied. Low or moderate inflation may be a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fan Chart (time Series)
In time series analysis, a fan chart is a chart that joins a simple line chart for observed past data, by showing ranges for possible values of future data together with a line showing a central estimate or most likely value for the future outcomes. As predictions become increasingly uncertain the further into the future one goes, these forecast ranges spread out, creating distinctive wedge or "fan" shapes, hence the term. Alternative forms of the chart can also include uncertainty for past data, such as preliminary data that is subject to revision. The term "fan chart" was coined by the Bank of England, which has been using these charts and this term since 1997 in its "Inflation Report" to describe its best prevision of future inflation to the general public. Fan charts have been used extensively in finance and monetary policy, for instance to represent forecasts of inflation. Implementation Predicted future values can be diagrammed in various ways; most simply, by a single p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayes Estimator
In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation. Definition Suppose an unknown parameter \theta is known to have a prior distribution \pi. Let \widehat = \widehat(x) be an estimator of \theta (based on some measurements ''x''), and let L(\theta,\widehat) be a loss function, such as squared error. The Bayes risk of \widehat is defined as E_\pi(L(\theta, \widehat)), where the expectation is taken over the probability distribution of \theta: this defines the risk function as a function of \widehat. An estimator \widehat is said to be a ''Bayes estimator'' if it minimizes the Bayes risk among all estimators. Equivalently, the estimator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is differentiable, the derivative test for finding maxima can be applied. In some cases, the first-order conditions of the likelihood function can be solved analytically; for instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when all observed outcomes are assumed to have Normal distributions with the same variance. From the perspective of Bayesian in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Component Analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. The principal components of a collection of points in a real coordinate space are a sequence of p unit vectors, where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bank Of England
The Bank of England is the central bank of the United Kingdom and the model on which most modern central banks have been based. Established in 1694 to act as the English Government's banker, and still one of the bankers for the Government of the United Kingdom, it is the world's eighth-oldest bank. It was privately owned by stockholders from its foundation in 1694 until it was nationalised in 1946 by the Attlee ministry. The Bank became an independent public organisation in 1998, wholly owned by the Treasury Solicitor on behalf of the government, with a mandate to support the economic policies of the government of the day, but independence in maintaining price stability. The Bank is one of eight banks authorised to issue banknotes in the United Kingdom, has a monopoly on the issue of banknotes in England and Wales, and regulates the issue of banknotes by commercial banks in Scotland and Northern Ireland. The Bank's Monetary Policy Committee has devolved responsibi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normalizing Constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. Definition In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function. Examples If we start from the simple Gaussian function p(x)=e^, \quad x\in(-\infty,\infty) we have the corresponding Gaussian integral \int_^\infty p(x) \, dx = \int_^\infty e^ \, dx = \sqrt, Now if we use the latter's reciprocal value as a normalizing constant for the former, defining a function \varphi(x) as \varphi(x) = \frac p(x) = \frac e^ so that its integral is unit \int_^\infty \varphi(x) \, dx = \int_^\infty \frac e^ \, dx = 1 then the function \varphi(x) is a probability d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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